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Beware of circles
I can't resist writing a methodological note about circles and wheels.
In the original posting by Michael, the wheel was conceived as a
Stating the obvious, and following up on Richard's note, for the
psychologist the circle can happen to be a concise and visually elegant
model of how humans organize a particular sensory/perceptual/cognitive
domain. Such a representation is often grounded in the construct of
similarity (objects that are close/far within the representation are
Now, my note is about those cases where circular representations are
extracted from the multidimensional scaling (MDS) analysis of behavioral
estimates of similarity. I have seen several of these.
What I have rarely seen pointed out is that popular MDS algorithms under
specific but not uncommon circumstances are prone to representing the
data as a circular (2D) or spherical (3D) structure, independently of
whether a circle/sphere is there or not. From my understanding, this MDS
modeling bias takes the name of "annular bias". A similar bias takes the
name of "horseshoe effect".
So, circles can be attractive, but there are times when we should beware
Bruno L. Giordano, Ph.D.
Postdoctoral Research Fellow
CIRMMT - Schulich School of Music
555 Sherbrooke Street West
Montréal, QC H3A1E3
+1 514 398 4535, Ext. 00900 (voice)
+1 514 398 2962 (fax)
Richard E Pastore wrote:
What is the reasonable goal of finding an auditory circle analogy to the
visual color circle?
Although the discussion has focused on finding an auditory analogue
to the color wheel, the discussion really has focused on the wheel.
Color is an artifical system that represents wavelength, with a
resulting representation of artificial "nonspectral hues" that do not
correspond to wavelengths in the visible spectrum. Color space was
defined using color mixing findings. The typical figure from those data
is essentially a triangle with rounded corners. The three primary
colors are at the three corners outside the space. If 400 and 700 nm
("blue" and "red") at maximum saturation are along the abscissa and
roughly 520 nm at the apex of the triangle, the wavelengths of the
visible spectrum runs along the outside from the 400 nm corner (blue)
through the apex (green) to the 700 nm corner (red). The abscissa maps
the non-spectral hues that are an artificial by-product of mixing the
long and short wavelengths. The space is populated from the edge to the
non-central region of total desaturation (white) by systematic
decreasing saturation or spectral purity. Complimentary colors are the
opposite ends of any line through "white." This 2-dimensional space is
actually a cross-section of a 3-D representation, with the 3rd dimension
being intensity or brightness.
The edge from the apex to the long wavelength corner (700 nm)
represents the opponent processing interaction between the long (red)
and middle (green) primaries. The space becomes populated with the
addition of the opponent processing between the short wavelength primary
(Blue) and the combined long and middle primaries (Yellow = Red +
Green). Because of the opponent neural coding that is driven by the
breakdown of the light sensitive photopigments, the afterimages are the
complimentary color of the original.
The "classic color circle" is a simplified, stylizied version of the
outer edge of the 2-D cross-section. It is round (a circle) that is
unpopulated and with a gap to represent the non-spectral hue portion of
color space - the circle is NOT complete.
Now, back to the original question that prompted the discussion:
What is the "auditory" circle intended to represent and in what way is
it analogous to the color circle?
Richard E Pastore
Distinguished Service Professor
Department of Psychology
Binghamton, NY 13902-6000
Office: (607) 777-2539